Bott periodicity and almost commuting matrices

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1 Bott periodicity and almost commuting matrices Rufus Willett January 7, 2019 Abstract We give a proof of the Bott periodicity theorem for topological K-theory of C -algebras based on Loring s treatment of Voiculescu s almost commuting matrices and Atiyah s rotation trick. We also explain how this relates to the Dirac operator on the circle; this uses Yu s localization algebra and an associated explicit formula for the pairing between the first K-homology and first K-theory groups of a (separable) C -algebra. 1 Introduction The aim of this note is to give a proof of Bott periodicity using Voiculescu s famous example [8] of almost commuting unitary matrices that cannot be approximated by exactly commuting unitary matrices. Indeed, in his thesis [4, Chapter I] Loring explained how the properties of Voiculescu s example can be seen as arising from Bott periodicity; this note explains how one can go backwards and use Loring s computations combined with Atiyah s rotation trick [1, Section 1] to prove Bott periodicity. Thus the existence of matrices with the properties in Voiculescu s example is in some sense equivalent to Bott periodicity. A secondary aim is to explain how to interpret the above in terms of Yu s localization algebra [10] and the Dirac operator on the circle. The localization algebra of a topological space X is a C -algebra C L pxq with the property that the K-theory of C L pxq is canonically isomorphic to the K-homology of X. We explain how Voiculescu s matrices, and the computations we need 1

2 to do with them, arise naturally from the Dirac and Bott operators on the circle using the localisation algebra. A key ingredient is a new explicit formula for the pairing between K-homology and K-theory in terms of Loring s machinery. We do not claim any real technical originality in this note: as will be obvious, our approach to Bott periodicity is heavily dependent on Loring s work in particular. However, we think this approach is particularly attractive and concrete it essentially reduces the proof of the Bott periodicity theorem to some formal observations and a finite dimensional matrix computation and hope that this exposition is worthwhile from that point of view. We also hope it is interesting insofar as it bridges a gap between different parts of the subject. A secondary motivation comes from possible applications in the setting of controlled K-theory; this will be explored elsewhere, however. Acknowledgment Thank you to the anonymous referee for several helpful comments. This work was partly supported by NSF grant DMS Preliminaries and Atiyah s rotation trick In this section, we set up notation and recall Atiyah s rotation trick. Let S 1 : tz P C z 1u. Let S denote the C -algebra C 0 ps 1 zt1uq, so S C 0 prq. For a C -algebra A, let SA denote the suspension S b A C 0 ps 1 zt1u, Aq. Let b P K 1 psq denote the Bott generator, i.e. b is the class of the unitary z ÞÑ z 1 in the unitization of S. Recall moreover that for a unital C -algebra A, the Bott map is defined on the class of a projection p P M n paq by β A : K 0 paq Ñ K 1 psaq, rps ÞÑ rbp ` p1 pqs, where the right hand side is understood to be the class in K 1 psaq of the unitary z ÞÑ z 1 p ` 1 p in the unitization of CpS 1 zt1u, M n paqq M n psaq. This identifies with an element in the unitary group of the nˆn matrices over the unitization of SA. One checks directly that the process above β A indeed gives a well-defined map on K-theory that induces a map in the non-unital case in the usual way, and moreover that the collection of maps is natural in 2

3 A in the sense that for any -homomorphism φ : A Ñ B, the corresponding diagram K 0 paq φ K 0 pbq β A β B K 1 psaq K 1 psbq pid S bφq commutes (see for example [7, Section ]). Theorem 2.1 (Bott periodicity theorem). For every C -algebra A, β A is an isomorphism. An important ingredient in our approach to this will be Atiyah s rotation trick [1, Section 1]: this allows one to reduce the proof of Bott periodicity to constructing a homomorphism α A : K 1 psaq Ñ K 0 paq for each C -algebra A, so that the collection tα A A a C -algebrau satisfies two natural axioms. As well as the fact that it simplifies the proof of Bott periodicity, a key virtue the rotation trick (as already pointed out in [1, Section 7]) is that the axioms satisfied by the α A are easily checked for several different natural analytic and geometric constructions of a collection of homomorphisms α A ; this paper essentially codifies the observation that Voiculescu s almost commuting matrices give yet another way of constructing an appropriate family tα A u. In order to give a precise statement of the axioms, recall from [3, Section 4.7] that for unital C -algebras A and B, the formulas ˆ : K 0 paq b K 0 pbq Ñ K 0 pa b Bq, rps b rqs ÞÑ rp b qs and ˆ : K 1 paq b K 0 pbq Ñ K 1 pa b Bq, rus b rps ÞÑ ru b p ` 1 b p1 pqs induce canonical external products on K-theory; moreover, applying these products to unitizations and restricting, these extend to well-defined products in the non-unital case too. Here then is a precise statement of the reduction that Atiyah s rotation trick allows: see the exposition in [3, Section 4.9] for a proof of the result as stated here. Proposition 2.2. Assume that for each C -algebra A there is homomorphism α A : K 1 psaq Ñ K 0 paq satisfying the following conditions. 3

4 (i) α C pbq 1. (ii) For any C -algebras A and B, the diagram below K 1 psaq b K 0 pbq α A b1 K 0 paq b K 0 pbq ˆ ˆ K 1 pspa b Bqq α AbB K 0 pa b Bq commutes; here the horizontal arrows are the canonical external products in K-theory discussed above, and we have used the canonical identifications SA b B ps b Aq b B S b pa b Bq SpA b Bq to make sense of the top horizontal arrow. Then α A and β A are mutually inverse for all C -algebras A. 3 Almost commuting matrices Our goal in this section is to construct homomorphisms α A : K 1 psaq Ñ K 0 paq with the properties in Proposition 2.2 above, and thus prove Bott periodicity. To motivate our construction, we start by recalling Voiculescu s almost commuting matrices. Let tδ 0,..., δ n 1 u be the canonical orthonormal basis for C n, and define unitaries by u n : δ k ÞÑ e 2πik{n δ k and v n : δ k ÞÑ δ k`1, (1) where the k`1 in the subscript above should be interpreted modulo n, or in other words v n : δ n 1 ÞÑ δ 0. It is straightforward to check that }ru n, v n s} Ñ 0 as n Ñ 8 (the norm here and throughout is the operator norm). Voiculescu [8] proved the following result. Theorem 3.1. There exists ɛ ą 0 such that if u 1 n, vn 1 are n ˆ n unitary matrices that actually commute, then we must have maxt}u n u 1 n}, }v n vn}u 1 ě ɛ for all n. 4

5 In words, the sequence of pairs pu n, v n q cannot be well-approximated by pairs that actually commute. Voiculescu s original proof of this fact uses nonquasidiagonality of the unilateral shift; there is a close connection of this to K-theory and Bott periodicity, but this is not obvious from the original proof. A more directly K-theoretic argument is due to Loring from his thesis [4]. Fix smooth functions f, g, h : S 1 Ñ r0, 1s with the properties in [4, pages 10-11]: the most salient of these are that f 2 ` g 2 ` h 2 f, fp1q 1, gp1q hp1q 0, and gh 0. For a pair of unitaries u, v in a unital C -algebra B, define the Loring element ˆ fpuq gpuq ` hpuqv epu, vq :, (2) v hpuq ` gpuq 1 fpuq which in general is a self-adjoint element of M 2 pbq. For later use, note that the conditions on f, g, h above imply the following formula, where for brevity we write e epu, vq, f fpuq, g gpuq, h hpuq, ˆhvg ` gv h e 2 e ` rf, hvs rv h, fs v h 2 v h 2 ; (3) in particular, if u and v happen to commute, this shows that e is an idempotent. Similarly, one can show that if }uv vu} is suitably small, then }e 2 e} is also small, so in particular the spectrum of e misses 1{2: indeed, this is done qualitatively in [4, Proposition 3.5], while a quantitative result for a specific choice of f, g, and h can be found in [5, Theorem 3.5]; the latter could be used to make the conditions on t that are implicit in our results more explicit. Thus if χ is the characteristic function of r1{2, 8q, then χ is continuous on the spectrum of e, and so χpeq is a well-defined projection in B. Loring shows that if e n P M 2n pcq is the Loring element associated to the matrices u n, v n P M n pcq as in line (1), then for all suitably large n, rankpe n q n 1. However, it is not difficult to see that if u n and v n were well-approximated by pairs of actually commuting matrices in the sense that the conclusion of Theorem 3.1 fails, then for all suitably large n one would have that rankpe n q 0. Thus Loring s work in particular reproves Theorem

6 Now, let us get back to constructing a family of homomorphisms α A : K 1 psaq Ñ K 0 paq satisfying the conditions of Proposition 2.2, and thus to prove Bott periodicity. For this, it will be convenient to have a continuously parametrised versions of the sequence pu n q, which we now build. Let then δ n : S 1 Ñ C be defined by δ n pxq? 1 2π z n, so the collection tδ n u npz is the canonical Fourier orthonormal basis for L 2 ps 1 q l 2 pzq. For each t P r1, 8q, define a unitary operator u t : L 2 ps 1 q Ñ L 2 ps 1 q to be diagonal with respect to this basis, and given by " e 2πint 1 δ u t δ n : n 0 ď t 1 n ď 1 otherwise δ n. Thus u t agrees with the operator of rotation by 2πt 1 radians on spantδ 0,..., δ n n ď tu, and with the identity elsewhere. Let A be a unital C -algebra faithfully represented on some Hilbert space H, and represent CpS 1 q on L 2 ps 1 q by multiplication operators in the canonical way. Represent SA tf P CpS 1, Aq fp1q 0u faithfully on L 2 ps 1, Hq as multiplication operators. Let χ be the characteristic function of tz P C Repzq ě 1u. 2 Note that the Bott element b acts on L 2 ps 1 q via the (backwards) bilateral shift. When compressed to the subspace spantδ 0,..., δ n n ď tu of L 2 ps 1 q, the operators u t and b are thus slight variants of Voiculescu s almost commuting unitaries from line (1). Lemma 3.2. Let A be a unital C -algebra, let Ă SA be the unitization of its suspension, and let v P Ă SA be a unitary operator. Then with notation as in the above discussion: (i) the spectrum of the Loring element epu t b 1 A, vq (cf. line (2)) does not contain 1{2 for all large t; (ii) the difference χpepu t b 1 A, vqq is in KpL 2 ps 1 qq b M 2 paq for all t; (iii) the function ˆ1 0 t ÞÑ χpepu t b 1 A, vqq is operator norm continuous for all large t. 6

7 Proof. For part (i), we first claim that for any f P CpS 1 q, rv t, fs Ñ 0 as t Ñ 8. Indeed, it suffices to check this for the Bott element bpxq z 1 as this function generates CpS 1 q as a C -algebra. With respect to the orthonormal basis tδ n : n P Zu we have that b acts by b : δ n ÞÑ δ n 1, i.e. b is the inverse of the usual bilateral shift. On the other hand, we have that }rv t, bs} }pv t bv t b qb} }v t bv t b }, and one computes directly that v t bv t b is a multiplication operator by an element of l 8 pzq that tends to zero as t tends to infinity, completing the proof of the claim. It follows from the claim that rv t b 1, f b as Ñ 0 as t Ñ 8 for any f P CpS 1 q, and any a P A. Hence rkpv t q b 1, as Ñ 0 for any a P CpS 1, Aq and any k P CpS 1 q (as kpv t q is in the C -algebra generated by v t ). Part (i) follows from this and the formula in line (3), plus the fact that hg gh 0. For part (ii), consider ˆ fpu epu t b 1 A, vq t q b 1 A gpu t q b 1 A ` phpu t q b 1 A qv. gpu t q b 1 A ` v phpu t q b 1 A q p1 fpu t qq b 1 A As e is norm continuous in the input unitaries, we may assume that v P CpS 1, Aq is of the form z ÞÑ ř M n M zn a n for some finite M P N. It follows from this, the formula for v t, and the facts that hp1q 0 gp1q and fp1q 1, that there exists N P N (depending on M and t) such that the operator epu t b 1 A, vq leaves some subspace of pl 2 ps 1 q b Hq 2 of the form pspantδ N,..., δ N u b Hq 2 ˆ1 0 invariant, and moreover that it agrees with the operator orthogonal complement of this subspace. It follows from this that ˆ1 0 χpepu t b 1 A, vqq on the is also zero on the orthogonal complement of pspantδ N,..., δ N u b Hq 2 7

8 and we are done. For part (iii), it is straightforward to check that the functions t ÞÑ hpu t q, t ÞÑ fpu t q and t ÞÑ gpu t q are continuous, as over a compact interval in the t variable, they only involve continuous changes on the span of finitely many of the eigenvectors tδ n u npz. It follows from this and the formula for epu, vq that the function t ÞÑ epu t b 1 A, vq is norm continuous. The claim follows from this, the fact that χ is continuous on the spectrum epu t b 1 A, vq for large t, and continuity of the functional calculus in the appropriate sense (see for example [7, Lemma 1.2.5]). Corollary 3.3. With notation as above, provisionally define by the formula for u P M n p r Aq α A : K 1 psaq Ñ K 0 pa b Kq rus ÞÑ rχpepv t b 1 MnpAq, uqqs j 1 0, where t is chosen sufficiently large (depending on u) so that all the conditions in Lemma 3.2 hold. Then this is a well-defined homomorphism for any C algebra A. Proof. This is straightforward to check from the formulas involved together with the universal property of K 1 as exposited in [7, Proposition 8.1.5], for example. Abusing notation slightly, we identity K 0 pa b Kq with K 0 paq via the canonical stabilization isomorphism, and thus treat α A as a homomorphism from K 1 psaq to K 0 paq. To complete our proof of Bott periodicity, it remains to check that these homomorphisms α A have the properties from Proposition 2.2. The second of these properties is almost immediate; we leave it to the reader to check. The first property, that α C pbq 1, is more substantial, and we give the proof here following computations in Loring s thesis. Proposition 3.4. With notation as above, α C pbq 1. 8

9 Proof. We must compute the element j 1 0 rχpepu t, bqqs P K 0 pkq Z for suitably large t, and show that it is one. We will work with an integer value t N for N suitably large. Note that with respect to the canonical basis tδ n u npz of L 2 ps 1 q, the element b acts as the (inverse of the) unilateral shift. On the other hand, on this basis u N pδ n q δ n for all n R p0, Nq. Define H N : spantδ n 1 ď n ď Nu, It follows from the above observations, the fact that fp1q 1 and hp1q gp1q 0, and a direct computation, that H 2 N is an invariant subspace of ˆ1 0 L 2 ps 1 q 2 for both epu N, bq, and for. Moreover, these two operators agree on the orthogonal complement ph 2 N qk. On H 2 N, epu N, bq agrees with the operator epu N, b N q, where we abuse notation by writing u N also for the restriction of u N to H N, and where b N : H N Ñ H N is the permutation operator defined by b N : δ n ÞÑ δ n`1 mod n (i.e. b N is the cyclic shift of the canonical basis). From these computations, we have that if we identify K 0 pkpl 2 ps 1 qqq Z and K 0 pbph N qq Z via the canonical inclusion BpH N q Ñ KpL 2 ps 1 qq (which induces an isomorphism on K-theory) then rχpepu N, bqqs j 1 0 rχpepu N, b N qqs j 1 0 P Z. Thus we have reduced the proposition (and therefore the proof of Bott periodicity) to a finite-dimensional matrix computation for N large: we must show that if e N : χpepu N, b N qq, then the trace of the 2N ˆ 2N matrix ˆ1 0 χpe N q is 1 for all large N, or equivalently, that the trace of the 2N ˆ 2N matrix e N is N 1 for all large N. This can be computing directly, following Loring. The computation is elementary, although slightly involved; we proceed as follows. 9

10 Step 1: }χpe N q e N } Op1{Nq for all N. For notational convenience, we fix N, drop the subscript N, and write f for fpv N q fpvq and similarly for g and h. We have that χpxq x ď 2 x 2 x for all x P R, whence from the functional calculus }χpeq e} ď 2}e e 2 }. Using the formula in line (3) above, it will thus suffice to show that the norm of ˆhbg ` gb h rf, bvs rb h, fs b h 2 b h 2 is bounded by C{N, where C ą 0 is an absolute constant not depending on N. Note that if a function k : S 1 Ñ C is Lipschitz with Lipschitz constant Lippkq, we have that }rk, bs} }bkb k} sup kpe 2πix q kp2πipx ` 1{Nqq ď 2πLippkq. xpr0,1s N This, combined with the fact that }h} ď 1 implies that ˆhrb, gs ` rg, b sh hrf, bs 4π ď plippfq ` Lippgq ` Lipphqq ru, fsh u phrh, bs ` rh, bshq N and we are done with this step. Step 2: trpχpe N qq trp3e 2 N 2e3 Nq Ñ 0 as N Ñ 8. The result of step one says that there is a constant C ą 0 such that the eigenvalues of e N are all within C{N of either one or zero for all large N. The function ppxq 3x 2 2x 3 has the property that pp0q 0, pp1q 1, and p 1 p0q p 1 p1q 0. Hence there is a constant D such that the eigenvalues of ppe N q are all within D{N 2 of either 0 or 1. It follows that }χpe N q ppe N q} ď D{N 2. Hence trpχpe N qq trpppe N qq trpχpe N q ppe N qq ď 2N}χpe N q ppe N q} ď 2D N. This tends to zero as N Ñ 8, completing the proof of step 2. Step 3: trpe 2 Nq N for all N. Using the formula in line (3) (with the same notational conventions used there) and rearranging a little, we get trpe 2 q trpeq ` trphbg ` gb hq ` trpb h 2 b h 2 q. 10

11 From the formula for e, we get that trpeq N. The second two terms are both zero using the trace property, and that gh 0. Step 4: trpe 3 N q pn 1 q Ñ 0 as N Ñ 8. 2 Again, we use the formula in line (3), multiplied by e e N to see that ˆhbg ` gb h trpe 3 q trpe 2 rf, hbs q ` tr e rb h, fs b hb h 2. The first term is N using step 3. Multiplying the second term out, simplifying using the trace properties and that gh 0, we see that the trace of the second term equals N 1 ÿ trp3h 2 pf bfb qq 3 hpe 2πik{N q 2 pfpe 2πik{N q fpe 2πipk 1q{N qq. k 0 Assuming (as we may) that f is differentiable, this converges as N tends to infinity to 3 ż 1 0 hpxq 2 f 1 pxqdx. Using that h 0 on r0, 1{2s, and that h f f 2 on r1{2, 1s (plus the precise form for f in [4, pages 10-11]) we get that 3 ż pfpe 2πix q fpe 2πix q 2 qf 1 pe 2πix qdx 3 ż 1 0 λ λ 2 dλ 1 2. This completes the proof of step 4. Combining steps 2, 3, and 4 completes the argument and we are done. 4 Connection with the localization algebra Homomorphisms α A with the properties required by Proposition 2.2 are maybe more usually defined using differential, or Toeplitz, operators associated to the circle. There is a direct connection between this picture and our unitaries u t, which we now explain; we will not give complete proofs here as this would substantially increase the length of the paper, but at least explain the key ideas and connections. The following definition is inspired by work of Yu [10] in the case that A is commutative. It agrees with Yu s original definition for unital commutative C -algebras. 11

12 Definition 4.1. Let A be a C -algebra, and assume that A is represented nondegenerately and essentially 1 on some Hilbert space H. Let C ub pr1, 8q, BpHqq denote the C -algebra of bounded uniformly continuous functions from r1, 8q to BpHq. The localization algebra of A is defined to be C LpAq : " f P Cub pr1, 8q, BpHqq fptqa P KpHq for all t P r1, 8q, a P A and }rfptq, as} Ñ 0 for all a P A If A C 0 pxq is commutative, we will write C L pxq instead of C L pc 0pXqq. The localization algebra of a C -algebra A does not depend on the choice of essential representation H up to non-canonical isomorphism, and its K- theory does not depend on H up to canonical isomorphism (these remarks follow from [2, Theorem 2.7]); thus we say the localization algebra of A, even thought this involves a slight abuse of terminology. Moreover, building on work of Yu [10] and Qiao-Roe [6] in the commutative case, [2, Theorem 4.5] gives a canonical isomorphism K paq Ñ K pc LpAqq from the K-homology groups of A to the K-theory groups of C L paq (at least when A is separable). One can define a pairing K i pc LpAqq b K j paq ÞÑ Z between the K-theory of the localization algebra (i.e. K-homology) and the K-theory of a C -algebra A. The most complicated case, and also the one relevant to the current discussion of this occurs when i j 1, so let us focus on this. Let pu t q tpr1,8q be a unitary in the unitization of C L paq and v in the unitization of A be another unitary (the construction also works with matrix algebras involved in exactly the same way, but we ignore this for notational simplicity). Let H be the Hilbert space used in the definition of the localization algebra, and for t P r1, 8q, let epu t, vq P M 2 pbphqq 1 This means that no non-zero element of A acts as a compact operator, so in particular, such a representation is faithful. *. 12

13 be the Loring element of line (2). One can check that for all large t, the spectrum of epu t, vq does not contain 1{2. Hence if χ be the characteristic function of r1{2, 8q, we get a difference ˆ1 0 χpepu t, vqq P M 2 pbphqq of projections for all suitably large t. It is moreover not difficult to check that this difference is in M 2 pkphqq, and thus defines an element in K 0 pkphqq Z, which does not depend on t for t suitably large. We may thus define j 1 0 xru t s, rvsy : repu t, vqs P K 0 pkq Z for any suitably large t. One checks that this formula gives a well-defined pairing between K 1 paq and K 1 paq. More substantially, one can check that it agrees with the canonical pairing between K-homology and K-theory (at least up to sign conventions). Let us go back to the case of interest for Bott periodicity. In terms of elliptic operators, one standard model for the canonical generator of the K- homology group K 1 ps 1 q of the circle is the class of the Dirac operator D i d, where θ is the angular coordinate. We consider D as an unbounded 2π dθ operator on L 2 ps 1 q with domain the smooth functions C 8 ps 1 q. Let χ : R Ñ r 1, 1s be any continuous function such that lim tñ 8 χptq 1, and for t P r1, 8q, define F t : χpt 1 Dq using the unbounded functional calculus. Concretely, each F t is diagonal with respect to the canonical orthonormal basis tδ n n P Zu, acting by F t : δ n ÞÑ χpt 1 nqδ n ; this follows as D : δ n ÞÑ nδ n for each n. Using the above concrete description of the eigenspace decomposition of F t, it is not too difficult to show that the function t ÞÑ F t defines an element of the multiplier algebra MpC L ps1 qq of the localization algebra C L ps1 q. Moreover, one checks similarly that the function t ÞÑ 1 2 pf t`1q maps to a projection in MpC L ps1 qq{c L ps1 q, and thus defines a class rds 0 P K 0 pmpc L ps1 qq{c L ps1 qq. Yu defines the K-homology class associated to this operator to be the image rds of rds 0 under the boundary map B : K 0 pmpc LpS 1 qq{c LpS 1 qq Ñ K 1 pc LpS 1 qq 13

14 (all this is part of a very general machine for turning elliptic differential operators into elements of K pc L ps1 qq: see [9, Chapter 8]). This boundary map is explicitly computable (compare for example [7, Section 12.2]): the image of rds 0 under this map is the class of the unitary e 2πi 1 2 pft`1q e πiχpt 1 Dq. Choosing χ to be the function which is negative one on p 8, 0s, one on r1, 8q, and that satisfies χptq 2t 1 on p0, 1q, we see that e πiχpt 1 Dq u t (4) i.e. the canonical generator of the K-homology group K 1 ps 1 q is given precisely by the class of u t in K 1 pc L ps1 qq. The formula for α A is then a specialization of the general formula above for the pairing, using the element of line (4) (more precisely, its amplification to CpS 1, Aq in an obvious sense). Thus our proof of Bott periodicity using almost commuting matrices fits very naturally into the localization picture of K-homology. References [1] M. Atiyah. Bott periodicity and the index of elliptic operators. Quart. J. Math. Oxford Ser. (2), 19(1): , [2] M. Dadarlat, R. Willett, and J. Wu. Localization C -algebras and K- theoretic duality. arxiv: , [3] N. Higson and J. Roe. Analytic K-homology. Oxford University Press, [4] T. Loring. The torus and noncommutative topology. PhD thesis, University of California, Berkeley, [5] T. Loring. Quantitative K-theory related to spin Chern numbers. Symmetry Integrability Geom. Methods Appl., 10, paper 077, [6] Y. Qiao and J. Roe. On the localization algebra of Guoliang Yu. Forum Math., 22(4): ,

15 [7] M. Rørdam, F. Larsen, and N. Laustsen. An Introduction to K-Theory for C -Algebras. Cambridge University Press, [8] D.-V. Voiculescu. Asymptotically commuting finite rank unitary operators without commuting apporximants. Acta Sci. Math. (Szeged), 45: , [9] R. Willett and G. Yu. Higher index theory. Available on the first author s website, [10] G. Yu. Localization algebras and the coarse Baum-Connes conjecture. K-theory, 11(4): ,